Reddit reviews Thinking Mathematically (2nd Edition)
We found 5 Reddit comments about Thinking Mathematically (2nd Edition). Here are the top ones, ranked by their Reddit score.
Reddit reviews Thinking Mathematically (2nd Edition)We found 5 Reddit comments about Thinking Mathematically (2nd Edition). Here are the top ones, ranked by their Reddit score.
I’m not sure Khan Academy is the most useful source; most of the assigned exercises I looked at a few years ago seemed pretty much trivial. You just watch someone solve a problem on a video, and then do exactly the same steps but with slightly different details. It’s an exercise in memory and copying, not in thinking for yourself. Basically the same curriculum as standard high school courses, just at your own pace. See Lockhart, “A Mathematician’s Lament” and Toom, “Word Problems in Russia and America”.
If you are self-studying the Gelfand and Kisilev books /u/TheBloodyNine1 mentioned are nice Russian books with some good problems in them, but also some text. If the text exposition is too fast or high level you could try reading the algebra and geometry books by Harold Jacobs. These have easier (standard American style) exercises but gentler exposition. If you are looking for medium to hard (by typical American standards) problems but also a good amount of step-by-step help with solving them, you might enjoy the Art of Problem Solving books, including those about algebra, geometry, basic number theory, “precalculus”.
Or for something a bit more poetic, check out Lockhart’s book Measurement.
The best way to learn the “why” of things in a real way is by doing the work for yourself. If someone just tries to tell you it won’t really sink in – you have to struggle with something for yourself before the explanation even has any relevance. Sometimes a book of nothing but problems can be just as useful as a book full of text.
See if you can work your way through problems such as those in Mathematical Circles (Russian Experience) (designed for ambitious Russian middle school students). Or you can look at the problems used at Exeter (famous private high school): Math 1, Math 2, Math 3–4, Math 4–5, Math 6, Discrete Math.
Or see if you can solve some past contest math problems. E.g. pick up a copy of a past AMC 12 (or AMC 10 or AMC 8 if those are too hard), and see how many problems you can do if you let yourself try to solve each one for 20 minutes without looking up the answer.
If you get through some of those and want less typical fare there are some fun topics in A Decade of the Berkeley Math Circle.
For some more general advice about problem solving methods (alongside problems), the book Thinking Mathematically is nice.
To be honest, the fastest way to improve is to find an expert tutor/mentor/coach to meet with face to face. Self-studying from books or websites or learning from class lectures and completely independent work is much more difficult / less efficient. There might be free tutoring resources available in your area if you hunt around (e.g. sometimes colleges will do free tutoring for nearby high school students).
Finally, if you get stuck on anything (problem, topic, ...) in particular, try /r/learnmath.
I'm also planning on doing a Masters in Math or CS. What do you plan to write for your masters?
> Anybody else feels like this?
I think its natural to doubt yourself, sometimes. I dont know what else to say, but just try to be objective and emotionless about it (when you get stuck in a problem).
The following books that helped me improve my math problem solving skills when I was an undergrad:
From Amazon, I see very good reviews for this book.
Also not a book, but a Stanford Coursera online course. Check it out here. What's great about this is it's free, you can (e-)meet and discuss with the professor and classmates, and there will certainly be mention of books and/or papers for reading more.
Best of luck to you! I'd be happy to hear from you in a few months about your progress, as I'm thinking about my career and the potential to get a PhD in something mathematical.
Yes! It looks similar - I'm only going by the course description though. Here's one of the texts from the course I took http://www.amazon.co.uk/Thinking-Mathematically-J-Mason/dp/0273728911
Check out Thinking Mathematically
and Betterexplained.